Multiplicative recurrence of M\"obius transformations

TL;DR

This paper characterizes multiplicative recurrence in images of positive integers under Möbius transformations, refutes a previous question, and extends Diophantine approximation results, confirming related conjectures.

Contribution

It provides a complete characterization of multiplicative recurrence for Möbius transformation images and extends Diophantine approximation results, addressing open questions in the field.

Findings

Complete characterization of multiplicative recurrence

Refutation of a previous open question

Extension and confirmation of conjectures in Diophantine approximation

Abstract

We establish a complete characterization of multiplicative recurrence for images of the positive integers under M\"obius transformations, answering a question of Donoso--Le--Moreira--Sun in the negative. As a consequence, we strengthen and extend a Diophantine approximation result of Charamaras--Mountakis--Tsinas, confirming their conjectures.